Parametric Equations Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Eliminate the parameter from x=2t+1x = 2t + 1 and y=t−3y = t - 3 to find the rectangular equation.

Solution

  1. 1
    Solve the xx-equation for tt: t=x−12t = \frac{x - 1}{2}.
  2. 2
    Substitute into the yy-equation: y=x−12−3y = \frac{x - 1}{2} - 3.
  3. 3
    Simplify: y=x−12−3=x−1−62=x−72y = \frac{x - 1}{2} - 3 = \frac{x - 1 - 6}{2} = \frac{x - 7}{2}.

Answer

y=x−72y = \frac{x - 7}{2}
Eliminating the parameter converts parametric equations back to a single rectangular equation. Solve one equation for tt and substitute into the other. The result here is a line with slope 12\frac{1}{2} and yy-intercept −72-\frac{7}{2}.

About Parametric Equations

A way of defining a curve by expressing both xx and yy as separate functions of a third variable (parameter), typically tt: x=f(t)x = f(t), y=g(t)y = g(t).

Learn more about Parametric Equations →

More Parametric Equations Examples

Example 2 medium

Eliminate the parameter from [formula] and [formula].

Example 3 medium

Find parametric equations for the line through [formula] and [formula].

Example 4 hard

Eliminate the parameter from [formula] and [formula] and describe the curve.

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